Determine whether the ancilla pull-out axiom can be localized

Determine whether the ancilla pull-out axiom (ca) for controllable quantum circuits with ancillas can be derived from the other axioms of \axanc, or replace it with a collection of more local axioms while retaining completeness.

Background

For controllable quantum circuits with ancillas, the axiom (ca) allows initialization and termination morphisms to be pulled out of a control. Unlike the axioms for controllable unitary circuits, (ca) involves an arbitrary number of qubits, so the authors do not obtain a bounded-locality axiomatization for the CPROP presentation.

The authors explicitly state that they believe (ca) cannot be directly derived from the remaining axioms of \axanc, while also acknowledging that replacing it with more local axioms may be possible. Thus, the independence of (ca) and the existence of a local replacement remain unresolved. Passing to CCPROPs removes the need to list (ca) separately because it follows from the conjugation law, but this does not resolve the corresponding question for the CPROP axiomatization.